Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space

WOS: 000378820200018We consider generalized Morrey spaces on quasi-metric measure spaces , in general unbounded, with variable exponent p(x) and a general function defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions . Our conditions do not suppose any assumption on monotonicity of in r.Science Development Foundation [EIF-2013-9(15)-46/10/1]; Presidium Azerbaijan National Academy of ScienceAzerbaijan National Academy of Sciences (ANAS); Russian Fund of Basic ResearchRussian Foundation for Basic Research (RFBR) [15-01-02732]; TUBITAK BIDEB Project [B.14.2.TBT.0.06.01.03.220.01-29104]The authors would like to express their gratitude to the referees for his (her) very valuable comments and suggestions. The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015. The research of S. Samko was partially supported by the Grant 15-01-02732 of Russian Fund of Basic Research and by the grant of TUBITAK BIDEB Project No.: B.14.2.TBT.0.06.01.03.220.01-29104 during his visit to Dumlupinar University, Turkey, in April of 2014